The \(K^+\rightarrow \pi ^+\nu \bar{\nu }\)  decay is a Flavour Changing Neutral Current
(FCNC) process that proceeds at the lowest order in the Standard Model
(SM) through electroweak box and penguin diagrams, both dominated by
\(t\) -quark exchange. The quadratic Glashow-Iliopoulos-Maiani (GIM)
mechanism and the transition from a top to a down quark make this
process extremely rare. Using tree-level elements of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix as external inputs, the SM
predicts the branching ratio to be BR
\( = (8.4\pm 1.0)\times 10^{-11}\)  [1]}, where the uncertainty
is dominated by those of the CKM parameters \(V_{cb}\)  and \(\gamma \) . The
intrinsic theoretical uncertainty is 3.6\(\%\) , related to the
uncertainty of the NLO (NNLO) QCD corrections to the top (charm) quark
contribution [2]}, [3]} and NLO electroweak
corrections [4]}. The uncertainties due to the hadronic
matrix element governing the \(K\) –\(\pi \)  transition are negligible,
because they are evaluated using the precisely measured branching
ratio of \(K^+\rightarrow \pi ^0e^+\nu \) , corrected for isospin-breaking and non-perturbative
effects calculated in [4]}, [6]}, [7]}.
